Calculation of Error between the Exact Solution and Solution of Parabolic Equation (Heat Equation) by Krylov Approximation Methods
Keywords:
Inverses Problems, heat Source, Krylov subspace, Matrix exponential, Krylov projection method, singularity of function, SVD method.Abstract
Having good estimates in the computation of the approximation to expressions for the form f(A)v is very important in practical applications if we know at what stage the algorithm has to stop i.e avoid the principle of "luckybreak". In this paper we develop an a posteriori upper bound on the Krylov subspace approximation error. We seek the error committed between the exact solution and solution of parabolic equations(heat equation) by Krylov approximation methods.The idea of the method is to approximate the action of the evolution operator on a given state vector by means projection process onto a Krylov subspace. This estimate will allow us not only to theoretically study the behavior of the convergence of the Krylov method as well as its stability but also allow us to give the exact size of the Krylov space according to the fixed stop test and the precisions Wish to establish.References
X.X.Li, F.Yang,The truncation method for identifying the heat source dependent on a spatial variable,
Computers and Mathematics with Applications 62 (2011), pp. 2497- 2505.
V. L. DRUSKIN and L. A. KNIZHNERMAN,Two Polynomial methods of calculating functions of
symmetric matrices, U.S.S.R comput.maths.math.phys., vol.29, No.6 (1989), pp. 112-121.
V. Druskin and L. Knizhnerman, Krylov subspace approximation of eigenpairs and matrix functions in
exact and computer arithmetic, Numer Linear Algebra Appl. 2 (1995), 205-217.
V. Druskin and L. Knizhnerman, Extended Krylov subspaces: approximation of the matrix square root
and related functions, SIAM J. Matrix Anal. Appl. 19 (1998), 755-771.
F.R Gantmacher, The Theory of Matrices, Volume one- Chelsea, New York, 1959.
Nicholas J. Higham , Matrix Functions, Theory and Application, SIAM - Philadelphia, USA, 2008.
Monaghan A.J , Chebyshev series approximation on complex domains, Durham theses, DurhamUniversity.
Available at Durham E-Theses Online: http://etheses.dur.ac.uk/7168/ , 1984.
Stefan Paszkowski , Computation applications of Chebyshev polynomials and series, Moscow, Nauka,
Y.Saad , Iterative Methods for Sparse Linear Systems, second ed., SIAM - Philadelphia, USA, 2003 .
E. B. Davies, Spectral Theory and Differential Operators, Cambridge University Press 1995.
J. Liesen, Z. Strakos, Krylov Subspace Methods: Principles and Analysis, Oxford University Press 2013.
Downloads
Published
How to Cite
Issue
Section
License
Authors who submit papers with this journal agree to the following terms.